On Rack Invariants Of Legendrian Knots

TL;DR

This paper introduces Legendrian rack invariants for oriented Legendrian knots, generalizing quandle invariants, and demonstrates their ability to detect certain geometric features like cusps.

Contribution

The paper defines Legendrian racks as a new invariant, proves their relation to quandle invariants, and uses automated theorem proving to discover their axioms.

Findings

Legendrian racks detect cusps in Legendrian knots

Automated theorem proving helped formulate the axioms

Legendrian racks generalize quandle invariants

Abstract

In this article, we introduce rack invariants of oriented Legendrian knots in the 3-dimensional Euclidean space endowed with the standard contact structure, which we call Legendrian racks. These invariants form a generalization of the quandle invariants of knots. These rack invariants do not result in a complete invariant, but detect some of the geometric properties such as cusps in a Legendrian knot. In the case of topologically trivial Legendrian knots, we test this family of invariants for its strengths and limitations. We further prove that these invariants form a natural generalization of the quandle invariant, by which we mean that any rack invariant under certain restrictions is equivalent to a Legendrian rack. The axioms of these racks are expressible in first order logic, and were discovered through a series of experiments using an automated theorem prover for first order logic. We also present the results from the experiments on Legendrian unknots involving auto-mated theorem provers, and describe how they led to our current formulation.